Line Integral

A line integral is an integral along a smooth curve C.

Scalar Line Integral

Given a smooth curve C, integrating a function f along C gives the scalar line integral. As the name implies, this returns a scalar value.

Parameterize f using r(t) for a ≤ t ≤ b to get:

This gives a straightforward calculation for arc length:

Vector Line Integral

A vector field F is defined for some domain D. Given a smooth curve C that exists entirely within D, the vector line integral is given by:

where T is the unit tangent vector.

Parameterize F using r(t) for a ≤ t ≤ b to get:

dr Reformulation

Another common notation follows from reformulating r'(t) as:

Therefore dr can be substituted into the above equation.

Again parameterize F using r(t) for a ≤ t ≤ b to get:

Conservative Reformulation

If vector field F is conservative, then there exists at least one function f satisfying F = ∇f. If given the start and end points A and B of the curve C, then:

Or parameterize f using r(t) for a ≤ t ≤ b to get:

Circular Integral

If C is closed and circular, i.e. there are no endpoints, the integral is notated like

and is generally evaluated using Green's theorem.

Green's Theorem

Circulation Form

Green's theorem enables conversion between double integrals and line integrals along a closed and circular curve C.

Consider a vector field expressed as F = <P(x,y), Q(x,y)> and a closed circular curve C parameterized as r(t) = <x(t), y(t)>. The circulation form of the theorem states that:

for a C that is oriented counter-clockwise. (For a clockwise oriented C, negate the formula.) In some cases, such an integral is expressed as being along the perimeter of a surface D rather than along curve C. In this case C is substituted for ∂D.

This formulation is particularly useful in cases where derivation eliminates all variables of P and Q, leaving an integral that is simply the area of region D (i.e., ∫∫_D dA) multiplied by some scalar.

This can also be expressed in terms of curl, leading to it sometimes being called the curl form.

(Note that k is the unit vector.)

Flux Form

The flux form or normal form of the theorem states that:

This can also be expressed in terms of divergence, leading to it sometimes being called the divergence form.

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